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人気のある三角関数 >

2-2sin^2(x/2)=sin^2(x)

解

2 - 2 sin^{2} (\frac{x}{2}) = sin^{2} (x)

解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = π + 2 πn

+1

度

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

解答ステップ

2 - 2 sin^{2} (\frac{x}{2}) = sin^{2} (x)

両辺から

sin^{2} (x)

を引く

2 - 2 sin^{2} (\frac{x}{2}) - sin^{2} (x) = 0

三角関数の公式を使用して書き換える

2 - sin^{2} (x) - 2 sin^{2} (\frac{x}{2})

ピタゴラスの公式を使用する:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= 2 - (1 - cos^{2} (x)) - 2 (1 - cos^{2} (\frac{x}{2}))

簡素化

2 - (1 - cos^{2} (x)) - 2 (1 - cos^{2} (\frac{x}{2})) : cos^{2} (x) + 2 cos^{2} (\frac{x}{2}) - 1

2 - (1 - cos^{2} (x)) - 2 (1 - cos^{2} (\frac{x}{2}))

- (1 - cos^{2} (x)) : - 1 + cos^{2} (x)

- (1 - cos^{2} (x))

括弧を分配する

= - (1) - (- cos^{2} (x))

マイナス・プラスの規則を適用する

- (- a) = a, - (a) = - a

= - 1 + cos^{2} (x)

= 2 - 1 + cos^{2} (x) - 2 (1 - cos^{2} (\frac{x}{2}))

拡張

- 2 (1 - cos^{2} (\frac{x}{2})) : - 2 + 2 cos^{2} (\frac{x}{2})

- 2 (1 - cos^{2} (\frac{x}{2}))

分配法則を適用する:

a (b - c) = ab - a c

a = - 2, b = 1, c = cos^{2} (\frac{x}{2})

= - 2 \cdot 1 - (- 2) cos^{2} (\frac{x}{2})

マイナス・プラスの規則を適用する

- (- a) = a

= - 2 \cdot 1 + 2 cos^{2} (\frac{x}{2})

数を乗じる:

2 \cdot 1 = 2

= - 2 + 2 cos^{2} (\frac{x}{2})

= 2 - 1 + cos^{2} (x) - 2 + 2 cos^{2} (\frac{x}{2})

簡素化

2 - 1 + cos^{2} (x) - 2 + 2 cos^{2} (\frac{x}{2}) : cos^{2} (x) + 2 cos^{2} (\frac{x}{2}) - 1

2 - 1 + cos^{2} (x) - 2 + 2 cos^{2} (\frac{x}{2})

条件のようなグループ

= cos^{2} (x) + 2 cos^{2} (\frac{x}{2}) + 2 - 1 - 2

数を足す/引く:

2 - 1 - 2 = - 1

= cos^{2} (x) + 2 cos^{2} (\frac{x}{2}) - 1

= cos^{2} (x) + 2 cos^{2} (\frac{x}{2}) - 1

= cos^{2} (x) + 2 cos^{2} (\frac{x}{2}) - 1

2倍角の公式を使用:

2 cos^{2} (x) - 1 = cos (2 x)

= cos^{2} (x) + cos (2 \cdot \frac{x}{2})

乗じる

2 \cdot \frac{x}{2} : x

2 \cdot \frac{x}{2}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{x \cdot 2}{2}

共通因数を約分する:

2

= x

= cos^{2} (x) + cos (x)

cos (x) + cos^{2} (x) = 0

置換で解く

cos (x) + cos^{2} (x) = 0

仮定:

cos (x) = u

u + u^{2} = 0

u + u^{2} = 0 : u = 0, u = - 1

u + u^{2} = 0

標準的な形式で書く

a x^{2} + b x + c = 0

u^{2} + u = 0

解くとthe二次式

u^{2} + u = 0

二次Equationの公式:

次の場合:

a = 1, b = 1, c = 0

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

1^{2} - 4 \cdot 1 \cdot 0 = 1

1^{2} - 4 \cdot 1 \cdot 0

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot 0

規則を適用

0 \cdot a = 0

= 1 - 0

数を引く:

1 - 0 = 1

= 1

規則を適用

1 = 1

= 1

u_{1, 2} = \frac{- 1 \pm 1}{2 \cdot 1}

解を分離する

u_{1} = \frac{- 1 + 1}{2 \cdot 1}, u_{2} = \frac{- 1 - 1}{2 \cdot 1}

u = \frac{- 1 + 1}{2 \cdot 1} : 0

\frac{- 1 + 1}{2 \cdot 1}

数を足す/引く:

- 1 + 1 = 0

= \frac{0}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{0}{2}

規則を適用

\frac{0}{a} = 0, a \neq = 0

= 0

u = \frac{- 1 - 1}{2 \cdot 1} : - 1

\frac{- 1 - 1}{2 \cdot 1}

数を引く:

- 1 - 1 = - 2

= \frac{- 2}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{- 2}{2}

分数の規則を適用する:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2}{2}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

u = 0, u = - 1

代用を戻す

u = cos (x)

cos (x) = 0, cos (x) = - 1

cos (x) = 0, cos (x) = - 1

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

以下の一般解

cos (x) = 0

cos (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

以下の一般解

cos (x) = - 1

cos (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

すべての解を組み合わせる

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = π + 2 πn

グラフ

人気の例

3cos(B)-1=5cos(B)+1

3 cos (B) - 1 = 5 cos (B) + 1

cos(θ)=(sqrt(3))/4

cos (θ) = \frac{3}{4}

cos (x) = \frac{- 3}{5}

solvefor x,sin(2x)=-1/2

so l v e f or x, sin (2 x) = - \frac{1}{2}

9cos(2θ)=27cos(θ)-18

9 cos (2 θ) = 27 cos (θ) - 18